Dr Reasoner is fond of square numbers and she is also fond of palindromic numbers (which read the same in both directions, for example 2638362, or 77200277). She is therefore particularly fond of numbers like 121 and 484, which are both palindromic and square.

However, she cannot find any 4 digit palindromic square numbers, and has conjectured that no such numbers exist.

Prove that her conjecture is correct without using technology (no calculators, computers or coding!)